1 Exam 1 Spring 2007.

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1 Exam Spring An object is moving along a line. At each time t, its velocity v(t is given by v(t = t 2 2 t 3. Find the total distance traveled by the object from time t = to time t = Use the method of substitution to evaluate each of the following and indicate what substitution you are using. If the integral is a definite integral, give an exact value and not a numerical approximation. sin x cos(cos x e ( + ln x 2 x + x 2 (c x 3. Suppose the derivative f of f is continuous, f( = 2, and What is the value of f(4, and why? 4 4. Evaluate each of the following expressions: d ( x 6 sin 2 x + 4e x ( d x t 6 sin 2 t + 4e t dt (c d 2 ( 8 4 x 6 sin 2 x + 4e x 5. Find the points at which the curves f (x = 7. y 2 = x and 3y x = 2 intersect. Then sketch the two curves and indicate the region enclosed between them. Calculate the area of that region enclosed between the two curves.

2 2 EXAM 2 SPRING The plane region R shown in the figure below is bounded by the curves y = x, y = 0, and x =. This region is rotated around the y-axis that is, the vertical axis (and not the x-axis! thereby generating a solid. Just express the volume V of that solid as an integral. (Do not evaluate that integral. 2 Exam 2 Spring Evaluate: ( x + e x + 3 sec 2 x 2. Evaluate: sin x + cos 2 x 3. Evaluate: ln(ln x x 4. Evaluate: x 2 cos x 5. Evaluate: 3x 9 x 2 6. Evaluate: tan 3 x cos x 7. Evaluate: x x 2 x 2 8. Determine whether each improper integral converges and, if it does, find its value. 0 x e x2

3 3 EXAM 3 SPRING (x 3 9. Determine whether each sequence {a n } n= converges and, if it does, find its limit. a n = 6n 2 5n + a n = 4 + ( n n Exam 3 Spring The parts of this question are not related. Use a relevant series to find the exact value of the repeating decimal 0.96 = as a rational number. For a certain series a n, its partial sums s n are given by: n= s n = 5n n + 4 Determine whether the series n= 2. Does the series converge? Why or why not? (n =, 2, 3,... a n converges and, if so, find its sum. (c n=2 n= n=2 n n ( n+ n4 + 2 n 4 + n 3 n (ln n 2 3. Approximate the sum of n= 3 ( n n 2 by using its first four terms. Round your answer to 4 decimal places (to the right of the decimal point. Find an upper bound on the error of the approximation that you obtained in. Justify why the method you use is appropriate.

4 4 FINAL EXAM SPRING Find the radius of convergence of the power series n= (x 3 n n 4 n. Now find the interval of convergence of the same power series. 5. Find the first 4 nonzero terms of the Taylor series for f(x = x ln x about x =. 6. Express Use to express + x 5 as the sum of a power series in x. Use notation as the sum of a series of numbers. + x5 /2 (c Use to approximate with an error less than x5 (Do not use your calculator to evaluate this integral! 4 Final Exam Spring The parts of this question are not related! Approximate the integral by using the Trapezoidal Rule with 0 + x3 n = 4 subintervals. Round your answer to 4 decimal places. Starting with the Maclaurin series for e x, find a power series expansion of x 2 e x. 2. The acceleration (in m/s 2 at time t of a particle moving along a straight line is given by a(t = 2t. Determinate the velocity v(t if v(0 = 2(m/s. Calculate the total distance the particle travels - not the displacement - over the time interval 0 t Use techniques of symbolic integration to evaluate: arcsin x (Hint: Try integration by parts. e x + 4. Find an equation of the tangent line at the point where t = to the curve having parametric equations { x = e 2t, y = t ln t.

5 4 FINAL EXAM SPRING Write parametric equations for the curve that has polar equation r = 3 cos θ: { x = y = (c Set up and evaluate a definite integral in order to find the area of the region enclosed by the polar curve r = 3 cos θ. 5. Set up (but do not yet evaluate an improper integral whose value would be the volume of the solid obtained by rotating around the x axis the region 0 y x, x. Now evaluate that improper integral. 6. Do the following series converge? Why or why not? n=0 n= 2 n + n ln n ln(n Express as the sum of a power series in x. Use -notation. +x 3 /2 Use to express as the sum of a series of numbers. Either use 0 + x3 - notation or else give at least the first five terms of the series.